Landscape family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings
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{{Technical data page}}
{{Technical data page}}


The '''landscape family''' of [[rank-3 temperament]]s tempers out the [[landscape comma]], 250047/250000.
The '''landscape family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[landscape comma]], 250047/250000.


Discussed elsewhere are:  
Discussed elsewhere are:  
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{{Mapping|legend=1| 3 0 0 4 | 0 1 0 -2 | 0 0 1 2 }}
{{Mapping|legend=1| 3 0 0 4 | 0 1 0 -2 | 0 0 1 2 }}
: mapping generators: ~63/50, ~3, ~5
: mapping generators: ~63/50, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.001{{c}}, ~3/2 = 701.938{{c}}, ~5/4 = 386.335{{c}}
* [[WE]]: ~63/50 = 400.0012{{c}}, ~3/2 = 701.9383{{c}}, ~5/4 = 386.3350{{c}}
* [[CWE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.939{{c}}, ~5/4 = 386.339{{c}}
: [[error map]]: {{val| +0.004 -0.013 +0.028 -0.021 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9387{{c}}, ~5/4 = 386.3385{{c}}
: error map: {{val| 0.000 -0.016 +0.025 -0.026 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: {{monzo list| 9 0 0 0 | 4 3 6 -3 | 0 0 9 0 | 4 -6 6 6 }}/9
: {{monzo list| 9 0 0 0 | 4 3 6 -3 | 0 0 9 0 | 4 -6 6 6 }}/9
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]
* [[9-odd-limit]]
: {{monzo list| 12 0 0 0 | 4 6 6 -3 | 0 0 12 0 | 8 -12 12 6 }}/12
: {{monzo list| 12 0 0 0 | 4 6 6 -3 | 0 0 12 0 | 8 -12 12 6 }}/12
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
: unchanged-interval (eigenmonzo) basis: 2.5.9/7


{{Optimal ET sequence|legend=1| 3, 12, 27, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555 }}
{{Optimal ET sequence|legend=1| 12, 27, 39d, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555, 22920d }}


[[Badness]] (Sintel): 0.094<!-- 0.02130 × 10<sup>-3</sup> by Smith's -->
[[Badness]] (Sintel): 0.0940


== Landy ==
== Landy ==
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{{Mapping|legend=1| 3 0 0 4 -1 | 0 1 0 -2 -2 | 0 0 1 2 3 }}
{{Mapping|legend=1| 3 0 0 4 -1 | 0 1 0 -2 -2 | 0 0 1 2 3 }}
: mapping generators: ~63/50, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.078{{c}}, ~3/2 = 701.696{{c}}, ~5/4 = 385.162{{c}}
* [[WE]]: ~63/50 = 400.078{{c}}, ~3/2 = 701.696{{c}}, ~5/4 = 385.162{{c}}
: [[error map]]: {{val| +0.234 -0.025 -0.684 -1.116 +1.632 }}
* [[CWE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.709{{c}}, ~5/4 = 385.341{{c}}
* [[CWE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.709{{c}}, ~5/4 = 385.341{{c}}
: error map: {{val| 0.000 -0.246 -0.972 -1.562 +1.287 }}


{{Optimal ET sequence|legend=1| 12, 15, 27e, 45e, 57, 60e, 72, 159 }}
{{Optimal ET sequence|legend=1| 12, 15, 27e, 45e, 57, 60e, 72, 159, 231, 330, 402d }}


[[Badness]] (Sintel): 0.792
[[Badness]] (Sintel): 0.792
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Comma list: 364/363, 441/440, 625/624
Comma list: 364/363, 441/440, 625/624


Mapping: {{Mapping| 3 0 0 4 -1 -12 | 0 1 0 -2 -2 -1 | 0 0 1 2 3 4 }}
Mapping: {{mapping| 3 0 0 4 -1 -12 | 0 1 0 -2 -2 -1 | 0 0 1 2 3 4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 400.055{{c}}, ~3/2 = 701.814{{c}}, ~5/4 = 385.398{{c}}
* WE: ~63/50 = 400.0548{{c}}, ~3/2 = 701.8141{{c}}, ~5/4 = 385.3981{{c}}
* CWE: ~63/50 = 400.000{{c}}, ~3/2 = 701.772{{c}}, ~5/4 = 385.442{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7720{{c}}, ~5/4 = 385.4420{{c}}


{{Optimal ET sequence|legend=0| 12f, 15, 27eff, 57f, 60eff, 72, 84, 87, 99ef, 159, 171 }}
{{Optimal ET sequence|legend=0| 12f, 15, 27eff, 45efff, 57f, 60eff, 72, 87, 159, 243, 330 }}


Badness (Sintel): 0.923
Badness (Sintel): 0.923
Line 75: Line 76:
[[Comma list]]: 250047/250000, 117440512/117406179
[[Comma list]]: 250047/250000, 117440512/117406179


{{Mapping|legend=1| 3 0 2 8 16 | 0 1 4 6 0 | 0 0 -5 -10 2 }}  
{{Mapping|legend=1| 3 0 -13 -22 10 | 0 1 4 6 0 | 0 0 5 10 -2 }}  
 
: mapping generators: ~63/50, ~3, ~10240/9801
: mapping generators: ~63/50, ~3, ~9801/5120


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.996{{c}}, ~3/2 = 701.960{{c}}, ~9801/5120 = 1124.286{{c}}
* [[WE]]: ~63/50 = 399.9958{{c}}, ~3/2 = 701.9604{{c}}, ~10240/9801 = 75.7013{{c}}
* [[CWE]]: ~63/50 = 400.000{{c}}, ~3/2 = 701.965{{c}}, ~9801/5120 = 1124.298{{c}}
: [[error map]]: {{val| -0.012 -0.007 +0.039 -0.034 +0.043 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.9652{{c}}, ~10240/9801 = 75.7024{{c}}
: error map: {{val| 0.000 +0.010 +0.059 -0.011 +0.087 }}


{{Optimal ET sequence|legend=1| 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee }}
{{Optimal ET sequence|legend=1| 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee }}


[[Badness]] (Sintel): 2.305
[[Badness]] (Sintel): 2.31


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Landscape family| ]] <!-- main article -->
[[Category:Landscape family| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 13:37, 19 April 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The landscape family of rank-3 temperaments tempers out the landscape comma, 250047/250000.

Discussed elsewhere are:

Considered below are landy and house, besides the 7-limit temperament.

Landscape

Subgroup: 2.3.5.7

Comma list: 250047/250000

Mapping: [⟨3 0 0 4], ⟨0 1 0 -2], ⟨0 0 1 2]]

mapping generators: ~63/50, ~3, ~5

Optimal tunings:

  • WE: ~63/50 = 400.0012 ¢, ~3/2 = 701.9383 ¢, ~5/4 = 386.3350 ¢
error map: ⟨+0.004 -0.013 +0.028 -0.021]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.9387 ¢, ~5/4 = 386.3385 ¢
error map: ⟨0.000 -0.016 +0.025 -0.026]

Minimax tuning:

[[9 0 0 0⟩, [4 3 6 -3⟩, [0 0 9 0⟩, [4 -6 6 6⟩]/9
unchanged-interval (eigenmonzo) basis: 2.5.7/3
[[12 0 0 0⟩, [4 6 6 -3⟩, [0 0 12 0⟩, [8 -12 12 6⟩]/12
unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 12, 27, 39d, 45, 57, 60, 72, 99, 171, 441, 612, 1236, 1407, 1578, 1848, 2019, 6498, 8346, 8517, 10536, 12555, 22920d

Badness (Sintel): 0.0940

Landy

Landy tempers out the werckisma and fantaresma, alongside the landscape comma.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356

Mapping: [⟨3 0 0 4 -1], ⟨0 1 0 -2 -2], ⟨0 0 1 2 3]]

Optimal tunings:

  • WE: ~63/50 = 400.078 ¢, ~3/2 = 701.696 ¢, ~5/4 = 385.162 ¢
error map: ⟨+0.234 -0.025 -0.684 -1.116 +1.632]
  • CWE: ~63/50 = 400.000 ¢, ~3/2 = 701.709 ¢, ~5/4 = 385.341 ¢
error map: ⟨0.000 -0.246 -0.972 -1.562 +1.287]

Optimal ET sequence: 12, 15, 27e, 45e, 57, 60e, 72, 159, 231, 330, 402d

Badness (Sintel): 0.792

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624

Mapping: [⟨3 0 0 4 -1 -12], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 2 3 4]]

Optimal tunings:

  • WE: ~63/50 = 400.0548 ¢, ~3/2 = 701.8141 ¢, ~5/4 = 385.3981 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.7720 ¢, ~5/4 = 385.4420 ¢

Optimal ET sequence: 12f, 15, 27eff, 45efff, 57f, 60eff, 72, 87, 159, 243, 330

Badness (Sintel): 0.923

House

House tempers out the quartisma alongside the landscape comma.

Subgroup: 2.3.5.7.11

Comma list: 250047/250000, 117440512/117406179

Mapping: [⟨3 0 -13 -22 10], ⟨0 1 4 6 0], ⟨0 0 5 10 -2]]

mapping generators: ~63/50, ~3, ~10240/9801

Optimal tunings:

  • WE: ~63/50 = 399.9958 ¢, ~3/2 = 701.9604 ¢, ~10240/9801 = 75.7013 ¢
error map: ⟨-0.012 -0.007 +0.039 -0.034 +0.043]
  • CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.9652 ¢, ~10240/9801 = 75.7024 ¢
error map: ⟨0.000 +0.010 +0.059 -0.011 +0.087]

Optimal ET sequence: 111, 159, 270, 696, 966, 1236, 1506, 2631, 2901, 4137, 4407, 8544e, 12951ce, 14187cee

Badness (Sintel): 2.31

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